Orthotropic elastic material inversion through PDE constrained optimization

Abstract

This paper addresses the problem of orthotropic material inversion from mechanical responses, with particular emphasis on the parametrization of the modulus tensor (matrix). Straightforward parametrization can lead to material instabilities (lack of positive definiteness of the modulus tensor), and failed forward solves during the optimization process. To address this, we present two approaches to impose positive-definiteness: (a) so-called Alpha parametrization that is physically transparent, and (b) parametrization based on Cholesky decomposition. Positive definiteness is implicitly satisfied for Cholesky parametrization, while an inequality constraint and some bound constraints are required for Alpha parametrization. The inequality constraint leads to increased computational cost, which is alleviated by a practical methodology of approximating the nonlinear constraint by the box constraints for practical applications. In the end, both parametrizations have similar computational cost. Numerical examples indicate that Cholesky parametrization may be better for fully orthotropic inversion, while Alpha parametrization may be better for the special cases of transversely isotropic and cubic materials